Given the coordinate matrix of $ x $ relative to a (nonstandard) basis $ B $ for $\mathbb{R}^n$, find the coordinate matrix of $ x $ relative to the standard basis.\[ B = \{(3, -1), (0, 1)\} \]\[[x]_B = \begin{bmatrix}3 \\5\end{bmatrix}\]\[ [x]_S = \begin{bmatrix}\hspace{1cm} \\\hspace{1cm}\end{bmatrix}\]Explanation:- The problem involves converting the coordinate matrix of a vector $ x $ from a given nonstandard basis $ B $ to the standard basis.- The basis $ B $ consists of the vectors $ (3, -1) $ and $ (0, 1) $.- The coordinate matrix of $ x $ relative to $ B $ is $\begin{bmatrix} 3 \\ 5 \end{bmatrix}$.- The expression $[x]_S$ represents the coordinate matrix of $ x $ relative to the standard basis, which needs to be determined.

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Given the coordinate matrix of x relative to a (nonstandard) basis 8 for R, find the coordinate matrix of x relative to the standard basis B-((3,-1), (0, 1)}, [x]s-

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

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Given the coordinate matrix of $ x $ relative to a (nonstandard) basis $ B $ for $\mathbb{R}^n$, find the coordinate matrix of $ x $ relative to the standard basis.

\[ B = \{(3, -1), (0, 1)\} \]

\[
[x]_B =
\begin{bmatrix}
3 \\
5
\end{bmatrix}
\]

\[ [x]_S =
\begin{bmatrix}
\hspace{1cm} \\
\hspace{1cm}
\end{bmatrix}
\]

Explanation:

- The problem involves converting the coordinate matrix of a vector $ x $ from a given nonstandard basis $ B $ to the standard basis.
- The basis $ B $ consists of the vectors $ (3, -1) $ and $ (0, 1) $.
- The coordinate matrix of $ x $ relative to $ B $ is $\begin{bmatrix} 3 \\ 5 \end{bmatrix}$.
- The expression $[x]_S$ represents the coordinate matrix of $ x $ relative to the standard basis, which needs to be determined.

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